On Containment for Linear Systems With Switching Topologies: A Novel State Transition Matrix Perspective

Area of Science:

• Control Theory
• Systems Engineering
• Networked Systems

Background:

• Investigates the complex containment control problem for multi-leader linear systems.
• Addresses challenges posed by arbitrary switching topologies and non-full-row rank input matrices.

Purpose of the Study:

• To develop a novel analysis framework for multi-leader containment control.
• To establish conditions for follower states to converge within the leaders' convex hull.
• To provide design methods for feedback gain matrices.

Main Methods:

• Utilizes a state transition matrix analysis framework.
• Connects multi-leader containment to a virtual leader-following system.
• Leverages properties of positive linear systems to derive conditions.

Main Results:

• Demonstrates that follower states converge to the convex hull of leader states.
• Identifies mild conditions for convergence based on positive linear system concepts.
• Illustrates conditions with a second-order linear system example.

Conclusions:

• The proposed framework effectively solves the multi-leader containment control problem under general conditions.
• The derived conditions are crucial for achieving desired state convergence.
• Two feedback gain matrix design methods are presented, requiring a constantly united spanning tree topology.